The common-sense view seems to involve a separately existing 'self', or 'Cartesian ego', to which our bodies and minds belong. This ego is the subject of conscious experiences. We can imagine magically finding ourselves in someone else's body. We may even be able to imagine having total amnesia and a sudden change in personality. So neither physical nor psychological continuity seems necessary for the folk conception of personal identity. All that matters is that it's me that has the conscious experiences.
I think this is one case where common-sense is utterly senseless. This picture posits a pure ego entirely independent of any physical or psychological facts. So how could we ever know about it? Our egos might be swapping bodies all the time. Perhaps five minutes ago I was you and you were me. My ego had your experiences, but then swapped bodies (entering 'Richard Chappell'), adopting my memories and forgetting all about yours. I seem to remember writing the previous paragraph five minutes ago, but perhaps it wasn't really me back then at all. Maybe you actually had that experience, and I just gained the memory when my ego jumped into this body more recently? How could we ever know?
Such talk of swapping bodies casts doubt on the very intelligibility of independently existing and undetectable egos. What does it even mean to say that you had the experiences in my memories? What sort of a 'you' is that, if you haven't been in any way influenced by the experienced event? (By that I mean that there is no causal connection between the event and your current self; it's as if it never happened. You don't have any memories of being Richard Chappell, your beliefs and behaviour are in no way affected, etc.) I don't think the claim has any sense at all! It's just so much gibberish masquerading as meaningful English.
But what's left if we dismiss the Cartesian pure ego? What am I, if not that fleeting spirit? Don't misunderstand me; I'm happy enough with the concept of 'me now'. I'm currently having conscious experiences (mostly consisting of various shades of puzzlement), and indeed am quite enjoying it. That much seems unproblematic. No, what I'm concerned about is not 'me now', but 'me then', and 'me later'. Will the Richard Chappell of ten minutes from now be the same person as I? He'll no doubt think he is [update - indeed I do!]; he'll have memories of being me, and so forth. But what if he's not? Perhaps 'I' am changing every single moment. Maybe each human life is made up of billions of consecutive persons, each owning the body and mind for a brief moment, before passing it on. It's a grim thought, really. I want to experience the future. I don't want to die and have some other 'self' usurp my mind and body, even if he mistakenly believes himself to be none other than I.
But I'm still thinking in Cartesian terms. I must stop that. Suppose instead there is no independent 'I', no subject or 'pure ego' independent of the experiences themselves. (This is awfully counterintuitive; I'm having trouble wrapping my - er, this - head around the idea.) Then such problems disappear. But it raises new ones: what is the connection between my temporal parts? What makes them together one whole? We can note a particular body's physical continuity through spacetime. We can also note the psychological continuity of its mind, in terms of overlapping memories and other such psychological connections between the temporal parts. Common sense suggests that personal identity is some further fact on top of these, but the problems mentioned above count against such a view. Perhaps talk of 'personal identity' really adds nothing new to the discussion. This is what Derek Parfit argues in Reasons and Persons (which I'm just halfway through - perhaps this explains my confusion!). He asks us to imagine a case of 'teletransportation', where his body is destroyed but an atom-for-atom replica is reconstructed elsewhere:
We could say that my Replica here will be me, or we could instead say that he will merely be someone else who is exactly like me. But we should not regard these as competing hypotheses about what will happen. For these to be competing hypotheses, my continued existence must involve a further fact. If my continued existence merely involves physical and psychological continuity, we know just what happens in this case. There will be some future person who will be physically exactly like me, and who will be fully psychologically continuous with me. This psychological continuity will have a reliable cause, the transmission of my blueprint. But this continuity will not have its normal cause, since this future person will not be physically continuous with me. This is a full description of the facts. There is no further fact about which we are ignorant. If personal identity does not involve a further fact, we should not believe that there are here two different possibilities: that my Replica will be me, or that he will be someone else who is merely like me. What could make these different possibilities? In what could the difference consist? (p.242, original emphasis)
'Me now' isn't going to have any other experiences, because 'me now' is stuck here in the present. (And look at that, he's gone already.) Very soon, someone physically and psychologically continuous with 'me now' is going to have some new experiences (and hopefully some sleep! Why oh why do I blog so late at night!?). That's it. End of story. Is that person really me? That simply depends on what we choose to make 'me' mean. We might as well stipulate the answer as 'yes', or our everyday language is going to run into a lot of problems. But there's nothing metaphysically significant going on here. There's no shared 'ego' who has both experiences. There's just two slices of spacetime that are connected by various physical and psychological relations. That's all I am, and you too.
Maybe it'll make more sense in the morning.